A graph is a mathematical object consisting of nodes connected by lines called edges. The nodes could represent for example people, and the edges could represent a connection from having slept together. Now, Megan has such a graph. Arguably, a graph that is symmetric is nicer than a regular one, which is why Megan suggests that they should sleep together.

The title text is a small-world joke on the concept of Erdős number. Paul Erdős was a Hungarian mathematician renowned for his eccentricity and productivity. He holds the world record for the number of published math papers, as well as for the number of collaborative papers. A person's Erdős number is the "collaborative distance" between the person and Erdős. Paul Erdős's Erdős number is 0 by definition. All of his 511 collaborators have the Erdős number 1; anyone who has collaborated on a mathematical or scientific paper with any of those collaborators has an Erdős number of 2, and so on. Thus, if you have written a paper with someone who's written a paper with someone who's written a paper with Paul Erdős, your Erdős number is 3. If you know a mathematician or are a mathematician you can calculate his/her/your Erdős number here.

The Collaboration Graph is the graph where each edge represents two people collaborated on a mathematical paper together, and the people represented are those with an Erdős number. Some of Erdős's colleagues have published papers about the properties of the Collaboration Graph, treating it as if it were a real mathematical object. One of these papers made the observation that the graph would have a certain very interesting property if two particular points had an edge between them. To make the Collaboration Graph have that property, the two disconnected mathematicians immediately got together, proved something trivial, and wrote up a joint paper. Explained here.

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